61. 62. 63. 64. X f(x) X f(x) X f(x) X 式 f(x) -1 0 1 2 0 224 -10 13 0224 -10 23 0224 1 2 3 224 -1 0
61. 62. 63. 64. X f(x) X f(x) X f(x) X 式 f(x) -1 0 1 2 0 224 -10 13 0224 -10 23 0224 1 2 3 224 -1 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Parts 61-64 please… please use the second image as reference for the work that needs to be done
Transcribed Image Text:**Problem IV:** Find a linear polynomial \( f(x) \) which is the best least squares fit to the following data.
**Data Sets:**
1. **Set 61:**
- \( x: -1, 0, 1, 2 \)
- \( f(x): 0, 2, 2, 4 \)
2. **Set 62:**
- \( x: -1, 0, 1, 3 \)
- \( f(x): 0, 2, 2, 4 \)
3. **Set 63:**
- \( x: -1, 0, 2, 3 \)
- \( f(x): 0, 2, 2, 4 \)
4. **Set 64:**
- \( x: -1, 1, 2, 3 \)
- \( f(x): 0, 2, 2, 4 \)
5. **Set 65:**
- \( x: -2, 0, 1, 2 \)
- \( f(x): 0, 2, 2, 4 \)
6. **Set 66:**
- \( x: -2, -1, 0, 2 \)
- \( f(x): 0, 2, 2, 4 \)
7. **Set 67:**
- \( x: -1, 0, 1, 2 \)
- \( f(x): 1, 2, 2, 3 \)
8. **Set 68:**
- \( x: -1, 0, 1, 3 \)
- \( f(x): 1, 2, 2, 3 \)
9. **Set 69:**
- \( x: -1, 0, 2, 3 \)
- \( f(x): 1, 2, 2, 3 \)
10. **Set 70:**
- \( x: -1, 1, 2, 3 \)
- \( f(x): 1, 2, 2, 3 \)
11. **Set 71:**
- \( x: -1, 0,
![**Problem 6.** Find a linear polynomial which is the best least squares fit to the following data:
\[
\begin{array}{c|ccccc}
x & -2 & -1 & 0 & 1 & 2 \\
\hline
f(x) & -3 & -2 & 1 & 2 & 5 \\
\end{array}
\]
We are looking for a function \( f(x) = c_1 + c_2 x \), where \( c_1, c_2 \) are unknown coefficients. The data of the problem give rise to an overdetermined system of linear equations in variables \( c_1 \) and \( c_2 \):
\[
\begin{cases}
c_1 - 2c_2 = -3, \\
c_1 - c_2 = -2, \\
c_1 = 1, \\
c_1 + c_2 = 2, \\
c_1 + 2c_2 = 5.
\end{cases}
\]
This system is inconsistent.
We can represent the system as a matrix equation \( Ac = y \), where:
\[
A = \begin{pmatrix}
1 & -2 \\
1 & -1 \\
1 & 0 \\
1 & 1 \\
1 & 2
\end{pmatrix}, \quad
c = \begin{pmatrix}
c_1 \\
c_2
\end{pmatrix}, \quad
y = \begin{pmatrix}
-3 \\
-2 \\
1 \\
2 \\
5
\end{pmatrix}.
\]
The least squares solution \( c \) of the above system is a solution of the normal system \( A^T Ac = A^T y \):
\[
\begin{pmatrix}
1 & 1 & 1 & 1 & 1 \\
-2 & -1 & 0 & 1 & 2
\end{pmatrix}
\begin{pmatrix}
1 & -2 \\
1 & -1 \\
1 & 0 \\
1 & 1 \\
1 & 2
\end{pmatrix}
\begin{pmatrix}
c_1 \\
c_2
\end{pmatrix}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b19493d-2e0b-4edb-977e-29215d3bb0dd%2Fae84c256-311c-44d4-ae03-3eb038d6825c%2F9poffl5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 6.** Find a linear polynomial which is the best least squares fit to the following data:
\[
\begin{array}{c|ccccc}
x & -2 & -1 & 0 & 1 & 2 \\
\hline
f(x) & -3 & -2 & 1 & 2 & 5 \\
\end{array}
\]
We are looking for a function \( f(x) = c_1 + c_2 x \), where \( c_1, c_2 \) are unknown coefficients. The data of the problem give rise to an overdetermined system of linear equations in variables \( c_1 \) and \( c_2 \):
\[
\begin{cases}
c_1 - 2c_2 = -3, \\
c_1 - c_2 = -2, \\
c_1 = 1, \\
c_1 + c_2 = 2, \\
c_1 + 2c_2 = 5.
\end{cases}
\]
This system is inconsistent.
We can represent the system as a matrix equation \( Ac = y \), where:
\[
A = \begin{pmatrix}
1 & -2 \\
1 & -1 \\
1 & 0 \\
1 & 1 \\
1 & 2
\end{pmatrix}, \quad
c = \begin{pmatrix}
c_1 \\
c_2
\end{pmatrix}, \quad
y = \begin{pmatrix}
-3 \\
-2 \\
1 \\
2 \\
5
\end{pmatrix}.
\]
The least squares solution \( c \) of the above system is a solution of the normal system \( A^T Ac = A^T y \):
\[
\begin{pmatrix}
1 & 1 & 1 & 1 & 1 \\
-2 & -1 & 0 & 1 & 2
\end{pmatrix}
\begin{pmatrix}
1 & -2 \\
1 & -1 \\
1 & 0 \\
1 & 1 \\
1 & 2
\end{pmatrix}
\begin{pmatrix}
c_1 \\
c_2
\end{pmatrix}
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